The optimum running approximation of a matrix filterbank by pseudo inverse matrices

Firstly, we establish a theorem showing that approximation of a matrix filterbank, which minimizes the average value of squared norms of error based on pseudo inverse matrices, is essentially equivalent to the optimum approximation which is derived by Kida that minimizes various worst-case measures of approximation-error at the same time. Secondly, under a restriction that Fourier-transforms of the input-matrices and analysis-matrices are expressed by Fourier series with finite terms having bounded norms, we present concrete procedure of deriving synthesis matrices of the optimum running approximation obtained by taking pseudo inverse matrices which are composed of constant coefficient matrices and sampling matrices of the matrix filterbank. The synthesis-filterbanks in the past optimum running approximation by Kida are given by combinations of block-wise synthesis-filterbanks defined in separate small domains in the variable domain. Besides, in these past contributions, the restriction of continuity is necessary with respect to the measure of error. Under the above restriction, we prove that these complex restrictions are not necessary. Thirdly, we present a reciprocal theorem with respect to exchange of analysis-matrices and synthesis-matrices in a matrix filterbank.